Computing a lower approximation of the compulsory part of a task with varying duration and varying resource consumption

This paper considers a generalisation of the classical RCPSP problem: the resource consumption of each task is continuously varying over time and the duration and the start of each task may vary within real intervals. A first contribution is a general model for describing the resource consumption of a task over time. This model is justified when considering continuously divisible resources. The second contribution is the computation of the compulsory part or core time of such a task. The compulsory part gives the task's resource consumption common to all feasible schedules. Hence, it can be used in a global resolution process such as constraint programming or branch and bound approaches. The presented polynomial algorithms use only two particular schedules of that task.